Aullhi Yusu , , Tukur Aulkir Sulimn , , , Evrn Hinl , Dumitru Blnu , ,

a Department of Computer Engineering, Biruni University, Istanbul, Turkey

b Department of Mathematics, Near East University TRNC, Mersin 10, Turkey

c Department of Mathematics, Federal University Dutse, Jigawa, Nigeria

d Department of Mathematics, Cankaya University, Ankara, Turkey

e Institute of Space Sciences, Magurele, Bucharest, Romania

f Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

ABSTRACT We solve the Ostrovsky equation in the absence of the rotation effect using the Hirota bilinear method and symbolic calculation.Some unique interaction phenomena have been obtained between lump solution, breather wave, periodic wave, kink soliton, and two-wave solutions.All the obtained solutions are validated by putting them into the original problem using the Wolfram Mathematica 12.The physical characteristics of the solutions have been visually represented to shed additional light on the acquired results.Furthermore, using the novel conservation theory, the conserved vectors of the governing equation have been generated.The discovered results are helpful in understanding particular physical phenomena in fluid dynamics as well as the dynamics of nonlinear higher dimensional wave fields in computational physics and ocean engineering and related disciplines.

Keywords:Lump solution New interaction phenomena Breather waves Symmetry analysis Conservation laws

1.Introduction

Lump solitons are utilized in a wide range of applied disciplines, including mathematics, chemistry, communication, biology,and, in particular, all aspects of engineering and physics [1-3]. Although some researchers used numerical simulation or analytical methods to research the output of such structures, the theoretical analysis for such systems needs to be further explored [4-6] .Rogue wave (RW) is an instinctive ocean waves that has based increasingly on the theoretical and experimental aspects [7] .In the simplest form, nonlinear Schrodinger equation’s RW was proposed in[8] .It is of worth noting that, wave phenomena in different areas,such as Bose-Einstein condensates, plasmas, nonlinear mechanics,biophysics and finance can be depicted [9-11] .

In order to create a new mix of functions using the Hirota bilinear apprach, several writers used lump solutions and their interaction phenomena to get some novel solutions.Literature has generated several important works on lump solutions.A number of lump techniques have been presented from various perspectives,including Zakharov [12] , lump wave solution [13] , and lump solution using the Hirota bilinear method [14-17] .When several important features of lump solutions are considered, it can be observed that solitons’ forms, amplitudes, and speeds are retained after collision with another soliton, which is the elastic property of collision.The interaction between the kink solitary wave and the rouge wave solution was also described in [18] .And many more[19-52] .

For every given partial differential equation, symmetries are transformations that make the whole space of the problem’s solutions invariant.Symmetries can be utilized to produce reductions and precise group-invariant solutions.Other analytical features, such as asymptotic and blow-up behavior, rely heavily on invariant solutions.Furthermore, explicit solutions may be utilized to verify the correctness and reliability of numerical solution methods.The Lie technique may be used to identify all acceptable Lie symmetries for a particular PDE.Furthermore, the conservation law of a particular equation of evolution is a continuity equation that gives basic values preserved for all solutions.Among other important uses, they allow the detection and construction of mappings to linear equations of nonlinear evolution equations.In addition, they can be used to test integrability[53-60] .

The interaction between internal waves and ocean topography has been an active field of research for long.Driving mechanism of waves solutions impacts the propagation of surface and internal gravity waves.These waves are quite common in oceans, lakes and the atmosphere, the mechanism of their derivation could play a significant part in ocean engineering.When waves enter areas of shallow water, they are influenced by the ocean floor.The water’s free orbital motion is interrupted, and water particles in orbital motion are no longer able to return to their original position.The swell gets bigger and steeper as the water gets shallower, eventually taking on the typical sharp-crested wave shape.After the wave breaks, it transforms into a tidal wave [61] .The model equation for the unidirectional propagation of weakly nonlinear long surface and internal waves of small amplitude in a rotating fluid is known as the Ostrovsky equation [62] and is given by

where χdenotes the free surface of the in-compressible and inviscid liquid and the parameter γmeasures the effect of rotation.When γ= 0 , Eq 1 changes to

Here, we employ the Hirota bilinear technique [63] to establish some RW, lump solutions and their interaction for the Ostrovsky equation appearing in Eq 2 .In addition, we will investigate its Lie symmetry, symmetry reduction, and conservation laws.Placing the ColeHopf transformation

into Eq 2 , we get the following bilinear form:

Using Eq 4 , we reach the lump, lump-soliton, lump-kink, lumpperiodic, breather wave and other interaction phenomena to Eq 2 that will be discussed in the next section.

2.Lump and its interaction phenomena

This section presents the lump and its interaction solutions to the Ostrovsky equation given in Eq 2 .

2.1.Lump solution

The lump solution to Eq 2 will be reported in this portion.Suppose that the positive quadratic solutions to Eq 4 to be

Substituting Eq 5 into Eq 4 , provides a polynomial in x and t.Collecting the coefficients of the same power, and equating each collection to zero, gives a system of equations.The values of the parameter are obtained by solving the obtained system of equations.Inserting the obtained values of the coefficients into Eq 3 , gives For

one reaches

Thus,

2.2.Lump-soliton solution

Herein, the lump-soliton to Eq 2 is reported.Consider the hyperbolic test function to be the solution of

Eq 4

where η= a 2 t + a 1 x + a 3 , ζ= a 5 t + a 4 x + a 6 , ξ= a 8 t + a 7 x + a 9 .Substituting Eq 8 into Eq 4 , we generates a polynomial in x , tand hyperbolic sine function.Collecting the same power coefficients,and equating each collection to zero, gives a system of equations.The values of the parameter are attained by solving the obtained system of equations.Inserting the obtained values into Eq 3 , gives For

Thus,

2.3.Lump-kink solution

Herein, the lump-kink solution to Eq 2 is provided.Taking into account the exponential test function as a solution to

Substituting Eq 11 into Eq 4 , produces a polynomial in x , tand exponential function.Collecting the same power coefficients, and equating each collection to zero, gives a system of equations.The values of the parameter are obtained by solving the obtained system of equations.Inserting the obtained values into Eq 3 , gives:For

we obtain

2.4.Lump-periodic solution

Fig.1.The (a) 3D profile at β= -6 .7 (b) 3D profile at β= 6 .7 of Eq 10 and (c) 3D profile at β= -1 .5 (d) 3D profile at β= 0 .4 of Eq 13 .

Herein, the lump-periodic solution to Eq 2 is given.Taking into account the trigonometric test function as a solution to Eq 4 Substituting Eq 14 into Eq 4 , produces a polynomial in x , tand trigonometric function.Collecting coefficients of the same power,and equating each collection to zero, gives a system of algebraic equations.The values of the coefficients are obtained by solving the obtained system equations.Inserting the obtained values of the coefficients into Eq 3 , gives: For

Thus,

Fig.2.The (a) 3D profile at β= -2 (b) 3D profile at β= 9 .02 (c) density profile at β= -2 (d) density profile at β= 9 .02 of Eq 16 .

2.5.Breather wave solutions

Herein, the breather wave solutions to Eq 2 is reported.Consider the following test function to be a trial solution to Eq 4 :

Substituting Eq 17 into Eq 4 , produces a polynomial in x , texponential and trigonometric functions.Collecting the coefficients of the same power, and equating each collection to zero, gives a system of equations.The values of the coefficients are obtained by solving the obtained system of equations.Inserting the obtained values of the coefficients into Eq 3 , gives: Case-1: For

Fig.3.The (a) 3D profile at β= -1 .2 (b) 3D profile at β= -0 .3 (c) density profile at β= -1 .2 (d) density profile at β= -0 .3 of Eq 19 .

we get

Thus,

Fig.4.The (a) 3D profile at β= -4 .3 (b) 3D profile at β= -0 .5 (c) density profile at β= -4 .3 (d) density profile at β= -0 .5 of Eq 21 .

Case-2:When

we have

Thus,

2.6.Some new interaction solutions

Here, we report some new interaction solutions to Eq 2 .Consider the following test function to be a trial solution to Eq 4 :

Plugging Eq 22 into Eq 4 , produces a polynomial in trigonometric,hyperbolic and exponential functions.Collecting the coefficients of the same power, and equating each collection to zero, we reach a system of equations.The values of the parameter are obtained by solving the obtained system of equations.Inserting the obtained values of the coefficients into Eq 3 , gives: Case-1: When

Thus,

Thus,

3.Symmetry analysis

Here we report the symmetry analysis, nonlinear selfadjointness and conservation laws for Eq 2 .For Eq 2 we have the vector fields as

Eq.Eq 2 admits the followings infinitesimals:

where F1(t) is an arbitrary function of t.And the symmetries are

3.1.Adjoint system and conditions for nonlinear self-adjointness

Theorem 3.1.A symmetry such as Lie point, Lie-B?cklund, nonlocal symmetry etc is given by

of a nonlinear partial differential equations

with an m dependent variables will have an adjoint equation

and Lagrangian given by

Considering Eq 2 , the formal Lagrangian can be given by

where υis the new-dependent variables called the nonlocal variables.The adjoint system can be obtained using

where

On the basis of Lagrangian reported in Eq 28 , one can get the adjoint equation as

Now, we want establish a differential substitution of the form

so that (2) will become nonlinear self-adjointness.To this aim, we insert Eq 40 into Eq 39 and by expressing χxtfrom (2) we reach

By equating to zero the coefficients of the derivatives χt, χx, χxx, χxxx, we attain

And the corresponding solution becomes

where c1and F1(t) are arbitrary constant and function of t, respectively.

3.2.Conservation laws

The fact that Eq 2 is a nonlinear self-adjointness, with the help of its point symmetries, we use the Noether operator N [29] -[31]to get (C1, C2) .The conserved vectors will satisfy

where

·The symmetry X1= ?tadmits the following conserved vectors:

4.Conclusion

It is a common knowledge that many science and engineering aspects that may be represented by nonlinear equations have an empirical parameters.Lump solutions therefore allow researchers the freedom to design and run experiments in order to decide certain parameters by creating convenient or natural conditions.In nonlinear sciences, therefore, analyzing and obtaining lump solutions is becoming more desirable.In this work, the Ostrovsky equation in the absence of rotational effect has been investigated by means of the Hirota bilinear approach and symbolic calculation.Ostrovsky equation is an equation for the unidirectional propagation of weakly nonlinear long surface and internal waves of small amplitude in a rotating fluid.As a consequence, various novel interaction phenomena between lump solution with two-wave, periodic wave, breather wave, periodic wave and kink solution have been acquired.A periodic wave has a wavelength and frequency determined by a repeating continuous pattern.Breathers are pulsating localized structures that have been used to mimic extreme waves in a variety of nonlinear dispersive media with a narrow banded starting process.Several recent investigations, on the other hand, imply that breathers can survive in more complex habitats,such as random seas, despite the attributed physical restrictions.

All the acquired solutions are verified by inserting them into the original equation with the help of the Wolfram Mathematica 12 package.The solution’s physical features were graphically depicted to shed more light on the results obtained (see Fig.1 , Fig.2 ,Fig.3 and Fig.4 ).In addition, the governing equation’s conserved vectors are developed using a new conservation theorem.The results obtained are useful in understanding the basic nonlinear scenarios in fluid dynamics as well as the dynamics related to computational physics and engineering sciences in nonlinear higher dimensional wave fields.Our future study will be on the bifurcation analysis and stochastic fractional solitons to the studied nonlinear model.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.